Ordering on the AMS Bookstore is limited to individuals for personal use only. Libraries and resellers, please contact cust-serv@ams.org for assistance. See our librarian page for additional eBook ordering options.

Well-Posedness for General \(2×2\) Systems of Conservation Laws

Share this page

Fabio Ancona; Andrea Marson

We consider the Cauchy problem for a strictly hyperbolic
\(2\times 2\) system of conservation laws in one space dimension \(
u_t+[F(u)]_x=0, u(0,x)=\bar u(x),\) which is neither linearly degenerate
nor genuinely non-linear. We make the following assumption on the
characteristic fields. If \(r_i(u), \ i=1,2,\) denotes the
\(i\)-th right eigenvector of \(DF(u)\) and
\(\lambda_i(u)\) the corresponding eigenvalue, then the set \(\{u :
\nabla \lambda_i \cdot r_i (u) = 0\}\) is a smooth curve in the
\(u\)-plane that is transversal to the vector field
\(r_i(u)\).

Systems of conservation laws that fulfill such assumptions
arise in studying elastodynamics or rigid heat conductors at low
temperature.

For such systems we prove the existence of a closed domain
\(\mathcal{D} \subset L^1,\) containing all functions with
sufficiently small total variation, and of a uniformly Lipschitz continuous
semigroup \(S:\mathcal{D} \times [0,+\infty)\rightarrow \mathcal{D}\)
with the following properties. Each trajectory \(t \mapsto S_t \bar u\)
of \(S\) is a weak solution of (1). Vice versa, if a piecewise Lipschitz,
entropic solution \(u= u(t,x)\) of (1) exists for \(t \in
[0,T],\) then it coincides with the trajectory of \(S\), i.e.
\(u(t,\cdot) = S_t \bar u.\)

This result yields the uniqueness and continuous dependence of
weak, entropy-admissible solutions of the Cauchy problem (1) with small initial
data, for systems satisfying the above assumption.

Readership

Graduate students and research mathematicians interested in
partial differential equations.

Title (HTML):
Well-Posedness for General \(2×2\) Systems of Conservation Laws

Author(s) (Product display):
Fabio Ancona;
Andrea Marson

Affiliation(s) (HTML):
University of Bologna, Bologna, Italy;
University of Padova, Padova, Italy

Abstract:

We consider the Cauchy problem for a strictly hyperbolic
\(2\times 2\) system of conservation laws in one space dimension \(
u_t+[F(u)]_x=0, u(0,x)=\bar u(x),\) which is neither linearly degenerate
nor genuinely non-linear. We make the following assumption on the
characteristic fields. If \(r_i(u), \ i=1,2,\) denotes the
\(i\)-th right eigenvector of \(DF(u)\) and
\(\lambda_i(u)\) the corresponding eigenvalue, then the set \(\{u :
\nabla \lambda_i \cdot r_i (u) = 0\}\) is a smooth curve in the
\(u\)-plane that is transversal to the vector field
\(r_i(u)\).

Systems of conservation laws that fulfill such assumptions
arise in studying elastodynamics or rigid heat conductors at low
temperature.

For such systems we prove the existence of a closed domain
\(\mathcal{D} \subset L^1,\) containing all functions with
sufficiently small total variation, and of a uniformly Lipschitz continuous
semigroup \(S:\mathcal{D} \times [0,+\infty)\rightarrow \mathcal{D}\)
with the following properties. Each trajectory \(t \mapsto S_t \bar u\)
of \(S\) is a weak solution of (1). Vice versa, if a piecewise Lipschitz,
entropic solution \(u= u(t,x)\) of (1) exists for \(t \in
[0,T],\) then it coincides with the trajectory of \(S\), i.e.
\(u(t,\cdot) = S_t \bar u.\)

This result yields the uniqueness and continuous dependence of
weak, entropy-admissible solutions of the Cauchy problem (1) with small initial
data, for systems satisfying the above assumption.